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A table of values is a simple method to understand how a function like a linear equation behaves. By choosing values for the variable, we can find the corresponding values of the function. Let's use the function \(f(x) = x - 4\) as an example. Start by selecting different values for \(x\), such as -2, -1, 0, 1, 2, 3, and 4. Substitute these \(x\) values into the function to determine \(f(x)\):

• For \(x = -2\), \(f(-2) = -2 - 4 = -6\)
• For \(x = 0\), simply, \(f(0) = 0 - 4 = -4\)
• For \(x = 4\), \(f(4) = 4 - 4 = 0\)

Creating such a table helps in visualizing how \(f(x)\) changes with \(x\) and provides specific points you can use for graphing.

The coordinate plane is a two-dimensional surface where we can plot points defined by pairs of numbers. This system consists of two perpendicular lines, called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. This setup allows us to locate any point using an ordered pair \((x, y)\). For example, the origin of a coordinate plane is at point \((0, 0)\). Each point in our table of values for \(f(x) = x - 4\) represents a unique location on the coordinate plane. You simply take each \((x, f(x))\) pair from your table and locate it on the plane. No matter the function, the coordinate plane remains a consistent method to visualize the relationships between variables.

Plotting points is one of the foundational skills in graphing functions. Using the table of values we created, each \((x, f(x))\) pair becomes a point on the graph. To plot these:

• Start with \((-2, -6)\). Begin at the origin \((0, 0)\), move 2 units left (for \(x = -2\)), then move 6 units down to place your first point.
• Next, for \((0, -4)\), stay on the y-axis and move down 4 units.
• Continue this for the remaining points using the same method.

Points on a linear equation like \(f(x) = x - 4\) will naturally align, showing the equation's constant rate of change, which should form a straight line once plotted and connected.

Linear equations represent straight lines when graphed on the coordinate plane. The equation \(f(x) = x - 4\) is typical of linear functions. It is in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m\) is 1, indicating the line rises one unit up for every unit it moves to the right. The y-intercept \(b\) is -4, which is where the line crosses the y-axis. This means when \(x = 0\), \(f(x) = -4\).
Linear equations are vital in understanding relationships where change is constant. As you plot and connect the points of such equations, the beautifully consistent pattern of a straight line appears, revealing the predictability and uniformity inherent in linear relationships.